Recurrence relations for harmonic and derangement numbers

Dae San Kim; Jongkyum Kwon; Kyo-Shin Hwang; Taekyun Kim

Recurrence relations for harmonic and derangement numbers

T0 review · 0 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper proves three finite-sum recurrences for derangement, harmonic, and degenerate harmonic numbers by comparing coefficients of bivariate generating functions.

desk verdict Three correct but modest recurrence identities for derangement, harmonic, and degenerate harmonic numbers; the math checks out, the conclusion oversells. read the letter →

arxiv 2509.10404 v1 pith:ACJOOPD4 submitted 2025-09-12 math.NT

Taekyun Kim , Dae san Kim , Jongkyum Kwon , Kyo-Shin Hwang This is my paper · ORCID

classification math.NT MSC 11B83

keywords recurrencerelationsharmonicnumbersderangementdegenerategeneratingfunctionshyperharmonicbinomialcoefficientselementarymethods

topics The Collatz Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's goal is to establish three new recurrence relations: one for derangement numbers, one for harmonic numbers, and one for degenerate harmonic numbers. Each recurrence expresses a value at index n+m as a finite weighted sum of lower-index values, with binomial coefficients as weights. The proofs are elementary, relying only on known generating functions and coefficient comparison rather than on advanced machinery. A sympathetic reader would take the central claim to be that these identities hold for all integers m,n≥0 and that they reveal a common structural pattern in sequences that are usually treated separately.

What carries the argument

The carrying mechanism is the exponential generating function for each sequence: 1/(1-t)e^{-t} for derangements, 1/(1-t) log(1/(1-t)) for harmonic numbers, and 1/(1-t) log_{-λ}(1/(1-t)) for degenerate harmonic numbers. The paper converts each into a bivariate series in x and y, factors it strategically, and extracts the coefficient of x^n y^m. The degenerate case additionally uses the product rule log_λ(xy)=log_λ(x)+x^λ log_λ(y) to make the factorisation work.

What would settle it

Check the recurrences for small explicit cases: for m=n=1, Theorem 2.2 gives binom(2,1)H_2 = H_0 + H_1 + H_1 binom(2,1), i.e. 3=3; Theorem 2.1 gives D_2 = 1. For the degenerate case, evaluate both sides of Theorem 2.3 at m=n=1 with λ=1/2 and compare the coefficient of xy obtained from (16) and (17); any mismatch disproves the recurrence.

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Extended reading notes

Core claim

For derangement numbers D_n, the paper proves that D_{m+n}/n! equals a double sum over l and k of binomial coefficients, a sign, and a factor k!/l! times D_l. For harmonic numbers H_n, it proves the binomial-weighted identity binom(m+n,m) H_{m+n} = sum_{k=0}^n H_k binom(m+n-k-1,n-k) + H_m binom(m+n,n), and the degenerate version replaces the last binomial's top by m+n-λ and uses degenerate harmonic numbers H_{n,λ}. All three recurrences are obtained by the same manoeuvre: take the known univariate exponential generating function, expand it as a bivariate series in x and y, factor it as a function of x times a function of y/(1-x), expand each factor, and compare coefficients of x^n y^m.

Load-bearing premise

The degenerate-harmonic theorem depends on the previously established generating function for degenerate harmonic numbers and on the product rule for degenerate logarithms; if either is invalid, or if the expansion of binom(m+n-λ,n) with a non-integer top is not legitimate, the degenerate recurrence does not follow.

Editorial extensions

If this is right

Combining Theorem 2.2 with the hyperharmonic identity (10) gives a clean formula for hyperharmonic numbers H_n^{(m+1)} as a sum of ordinary harmonic numbers: H_n^{(m+1)} = sum_{k=0}^n H_k binom(m+n-k-1,n-k).
The derangement recurrence yields a finite double-sum expression for D_{m+n} in terms of lower derangement numbers, which can serve as a computational shortcut and as a basis for congruence arguments.
The degenerate harmonic recurrence reduces to the ordinary harmonic recurrence in the limit λ→0, so the identities are consistent with the classical specialisation.
Combining Theorem 2.3 with relation (13) produces an explicit formula for degenerate hyperharmonic numbers in terms of degenerate harmonic numbers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same bivariate coefficient-comparison strategy should work for any sequence whose generating function is a rational function times a logarithm or exponential of the same shape; this extension is not claimed in the paper itself.
Setting m=1 in the harmonic recurrence collapses to the standard H_{n+1}=H_n+1/(n+1) after simplification, so the general identity can be read as a systematic refinement of the basic defining relation.
For positive integer λ, the degenerate binomial top m+n-λ is an ordinary integer, and the degenerate recurrence might specialise to combinatorial identities that could be tested against known finite-difference or q-analogue results; the paper does not explore this.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper proves three recurrence identities that appear to be new: a double sum for derangement numbers, a single-sum identity for harmonic numbers, and a degenerate-parameter version for degenerate harmonic numbers. The derivations are elementary coefficient comparisons on known generating functions, in the same style the authors have used before. I checked the coefficient manipulations and they are valid. The stress-test concern about Theorem 2.3 is not a real problem: identity (8) follows immediately from the definition (7), and the product rule (6) is a purely algebraic identity for formal binomial series. The non-integer binomial coefficient binom(m+n−λ, n) is a polynomial in λ and the comparison in Q[λ][[x,y]] is legitimate. So the central claims stand.

What is genuinely good: the identities are correctly derived, they are not in the cited references (as far as I can tell), and the paper is short and transparent. The method is standard, but it is honestly executed. The harmonic number identity in Theorem 2.2 is clean and connects to hyperharmonic numbers through (10), and the degenerate analogue is a reasonable extension.

Soft spots: the Conclusion overstates what has been done. Calling these results a testament to elementary techniques uncovering profound truths is too much for a set of coefficient comparisons. The abstract promises potential applications but none are given or discussed. There is also no attempt to relate these recurrences to known identities beyond the immediate hyperharmonic corollaries. The final display in the degenerate case divides by binom(λ−1, m), which needs a nonzero restriction or a limiting interpretation; this is minor and does not affect Theorems 2.1–2.3. There are a few typesetting typos in the double series, but they do not obscure the argument.

Self-citation is not an issue here: the cited degenerate generating function really is the authors' own earlier work, and it is correct. The novelty is real but modest; these are straightforward consequences of known generating functions. A reader who works on degenerate special numbers or collects harmonic/derangement identities will find this useful. The paper is not a major advance, but it is a valid, correct contribution of limited scope.

For peer review: I would send it to a referee rather than desk-reject, because the math is correct and the identities are new. Ask the authors to trim the conclusion, add a note about the nonzero restriction, and perhaps survey the literature more carefully for prior statements of the harmonic identity. If the journal has a very low tolerance for incremental results, a desk rejection is defensible, but the work deserves a fair reading.

Referee Report

0 major / 5 minor

Summary. The paper derives three explicit recurrence relations: Theorem 2.1 for derangement numbers, Theorem 2.2 for harmonic numbers, and Theorem 2.3 for degenerate harmonic numbers. The proofs use bivariate generating functions and coefficient extraction: each left-hand side is written as a Taylor shift F(x+y), then decomposed using exponential, ordinary, or degenerate logarithmic identities, expanded in double series, and coefficients are compared. A supplementary formula for degenerate hyperharmonic numbers is displayed after Theorem 2.3.

Significance. The recurrences appear correct, and the derivations are fully explicit and verifiable. The strength of the paper is its elementary, self-contained style: the key coefficient manipulations are legitimate formal power series identities. The degenerate case, while depending on definitions from prior work, is handled carefully via the algebraic product rule (6) and the generating function (8). However, the contribution is incremental: the recurrences are direct consequences of known generating functions, and the conclusion overstates their novelty and significance. If the journal welcomes elementary recurrence papers of this type, the manuscript is acceptable after minor revision.

minor comments (5)

[After Theorem 2.3] The unnumbered display divides by binom(λ−1,m). For values of λ with binom(λ−1,m)=0 (e.g., λ=1, m≥1), the expression is undefined. The identity should either be restricted to parameters with binom(λ−1,m)≠0 or be stated as an equality in the field of rational functions in λ. This is a local fix and does not affect Theorems 2.1–2.3.
[Eq. (8)] The derivation of the generating function (8) from (7) is asserted but not shown. A one-line verification using the expansion of log_{−λ}(1/(1−t)) would make the paper self-contained, particularly since the sign convention (log_{−λ}) is unusual.
[Section 3] The conclusion's language ('profound mathematical truths', 'new and structured way') overstates the contribution; the recurrences are obtained by standard generating-function manipulations. Please rephrase to a more measured assessment.
[Section 2, Eq. (14)] The passage from ordinary power series to the exponential basis (x^n/n!) in (14) is not explained; a brief note on this coefficient conversion would improve readability.
[Title/Header] The first-page header reads 'RECURRENCE RELA TIONS'; please correct. Throughout the text, some equations have irregular spacing; a careful proofreading pass is recommended.

Circularity Check

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No circularity: the recurrences follow from elementary generating-function manipulations; the cited degenerate-harmonic facts are either re-derived in the text or are immediate from the definitions.

full rationale

The central claims (Theorems 2.1, 2.2, 2.3) are not assumed inputs of the derivations. Theorem 2.1 extracts the derangement recurrence from coefficient comparison in (14), which starts from the standard generating function (2). Theorem 2.2 does the same for harmonic numbers from (4). Although (4) is accompanied by citations [7,8,15], the paper states it as 'From (3), we have', and (3) is the definition H_n=1+...+1/n, so (4) is immediate: the coefficient of t^n in (1/(1-t)) log(1/(1-t)) is exactly ∑_{k=1}^n 1/k = H_n. Theorem 2.3 uses (6) and (8). Equation (8) is not an unverified imported assumption: the text says 'From (7), we derive the generating function of degenerate harmonic numbers: (8)', and (7) is the definition of H_{n,λ}; expanding log_{-λ}(1/(1-t)) = (1/λ)∑_{k≥1} λ choose k (-1)^{k-1} t^k and multiplying by 1/(1-t) gives exactly the partial sums in (7). Equation (6) is an algebraic consequence of (5). Thus the coefficient identity in (17) is a legitimate comparison of two expansions of the same formal power series. The later uses of (10) and (13) occur after Theorems 2.2 and 2.3 and only express corollaries, so they are not load-bearing for the recurrences. Self-citations appear, but they are not used as a substitute for proof; the recalled facts are either standard or re-derived in the paper itself. No reduction of a stated conclusion to its own input can be exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameters are fitted. The paper relies on standard generating functions for derangement and harmonic numbers and on the authors' previously defined degenerate harmonic numbers and degenerate logarithm; these are domain assumptions from the cited literature.

assumptions (4)

domain assumption Generating function for derangement numbers: e^{-t}/(1-t) = sum D_n t^n/n!
Quoted from references [3,9]; standard and independent of this paper.
domain assumption Generating function for harmonic numbers: (1/(1-t)) log(1/(1-t)) = sum H_n t^n
Quoted from references [7,8,15]; standard and derivable from the definition of H_n.
domain assumption Generating function for degenerate harmonic numbers: (1/(1-t)) log_{-lambda}(1/(1-t)) = sum H_{n,lambda} t^n
From the authors' prior work [7,8,15]; derivable from definition (7) but not independently verified in this paper.
domain assumption Degenerate logarithm product rule: log_lambda(xy) = log_lambda(x) + x^lambda log_lambda(y)
Stated as equation (6) and follows directly from the definition log_lambda(t) = (t^lambda - 1)/lambda.

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Cite this review

Pith. "Pith review of Recurrence relations for harmonic and derangement numbers." pith.science (2026). https://pith.science/paper/ACJOOPD4

@misc{pith2026250910404,
  author       = {Pith},
  title        = {Pith review of: Recurrence relations for harmonic and derangement numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ACJOOPD4}},
  note         = {Machine review of arXiv:2509.10404}
}

read the original abstract

We use elementary methods to establish three key recurrence relations: one for derangement numbers, a second for harmonic numbers, and a third for degenerate harmonic numbers. Our results not only contribute to the understanding of the underlying structure of these numbers but also highlight the effectiveness of elementary techniques in discovering new mathematical properties. The findings have potential applications in various fields where these numbers appear, including combinatorics, probability, and computer science.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

S.; Acikgoz, M.; Araci, S.A new construction on the degenerate Hurwitz-zeta function associated with certain applications,Proc

Aydin, M. S.; Acikgoz, M.; Araci, S.A new construction on the degenerate Hurwitz-zeta function associated with certain applications,Proc. Jangjeon Math. Soc.25(2022), no. 2, 195-203

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Benjamin, A. T.; Preston, G. O.; Quinn, J. J.A Stirling encounter with harmonic numbers. Math. Mag.75(2002), no. 2, 95-103

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S.Spivey-type recurrence relations for degenerate Bell and Dowling polyno- mials,Russ

Kim, T.; Kim, D. S.Spivey-type recurrence relations for degenerate Bell and Dowling polyno- mials,Russ. J. Math. Phys.32(2025), no. 2, 288-296

2025

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S.Combinatorial identities involving degenerate harmonic and hyperharmonic numbers,Adv

Kim, T.; Kim, D. S.Combinatorial identities involving degenerate harmonic and hyperharmonic numbers,Adv. Appl. Math.148(2023), Paper No. 102535, 15 pp

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S.Some identities involving degenerate Stirling numbers associated with sev- eral degenerate polynomials and numbers,Russ

Kim, T.; Kim, D. S.Some identities involving degenerate Stirling numbers associated with sev- eral degenerate polynomials and numbers,Russ. J. Math. Phys.30(2023), no. 1, 62-75

2023